Difference between revisions of "User:Jon Awbrey/MNO"

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{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f0f0ff; font-weight:bold; text-align:center; width:80%"
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|+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math>
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| width="20%" | <math>\mathcal{L}_1</math>
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| width="20%" | <math>\mathcal{L}_2</math>
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| width="20%" | <math>\mathcal{L}_3</math>
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| width="20%" | <math>\mathcal{L}_4</math>
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|-
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| Decimal
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| Binary
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| Sequential
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| Parenthetical
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|-
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| &nbsp;
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| align="right" | <math>p =\!</math>
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| 1 1 1 1 0 0 0 0
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| &nbsp;
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|-
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| &nbsp;
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| align="right" | <math>q =\!</math>
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| 1 1 0 0 1 1 0 0
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| &nbsp;
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|-
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| &nbsp;
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| align="right" | <math>r =\!</math>
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| 1 0 1 0 1 0 1 0
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| &nbsp;
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|}
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:80%"
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|-
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| width="20%" | <math>f_{104}\!</math>
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| width="20%" | <math>f_{01101000}\!</math>
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| width="20%" | 0 1 1 0 1 0 0 0
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| width="20%" | <math>( p , q , r )\!</math>
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|-
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| <math>f_{148}\!</math>
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| <math>f_{10010100}\!</math>
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| 1 0 0 1 0 1 0 0
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| <math>( p , q , (r))\!</math>
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|-
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| <math>f_{146}\!</math>
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| <math>f_{10010010}\!</math>
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| 1 0 0 1 0 0 1 0
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| <math>( p , (q), r )\!</math>
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|-
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| <math>f_{97}\!</math>
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| <math>f_{01100001}\!</math>
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| 0 1 1 0 0 0 0 1
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| <math>( p , (q), (r))\!</math>
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|-
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| <math>f_{134}\!</math>
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| <math>f_{10000110}\!</math>
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| 1 0 0 0 0 1 1 0
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| <math>((p), q , r )\!</math>
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|-
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| <math>f_{73}\!</math>
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| <math>f_{01001001}\!</math>
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| 0 1 0 0 1 0 0 1
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| <math>((p), q , (r))\!</math>
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|-
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| <math>f_{41}\!</math>
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| <math>f_{00101001}\!</math>
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| 0 0 1 0 1 0 0 1
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| <math>((p), (q), r )\!</math>
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|-
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| <math>f_{22}\!</math>
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| <math>f_{00010110}\!</math>
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| 0 0 0 1 0 1 1 0
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| <math>((p), (q), (r))\!</math>
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|}
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{|  align="center" border="1" cellpadding="4" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:80%"
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|-
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| width="20%" | <math>f_{233}\!</math>
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| width="20%" | <math>f_{11101001}\!</math>
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| width="20%" | 1 1 1 0 1 0 0 1
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| width="20%" | <math>(((p), (q), (r)))\!</math>
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|-
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| <math>f_{214}\!</math>
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| <math>f_{11010110}\!</math>
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| 1 1 0 1 0 1 1 0
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| <math>(((p), (q), r ))\!</math>
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|-
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| <math>f_{182}\!</math>
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| <math>f_{10110110}\!</math>
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| 1 0 1 1 0 1 1 0
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| <math>(((p), q , (r)))\!</math>
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|-
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| <math>f_{121}\!</math>
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| <math>f_{01111001}\!</math>
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| 0 1 1 1 1 0 0 1
 +
| <math>(((p), q , r ))\!</math>
 +
|-
 +
| <math>f_{158}\!</math>
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| <math>f_{10011110}\!</math>
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| 1 0 0 1 1 1 1 0
 +
| <math>(( p , (q), (r)))\!</math>
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|-
 +
| <math>f_{109}\!</math>
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| <math>f_{01101101}\!</math>
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| 0 1 1 0 1 1 0 1
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| <math>(( p , (q), r ))\!</math>
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|-
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| <math>f_{107}\!</math>
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| <math>f_{01101011}\!</math>
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| 0 1 1 0 1 0 1 1
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| <math>(( p , q , (r)))\!</math>
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|-
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| <math>f_{151}\!</math>
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| <math>f_{10010111}\!</math>
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| 1 0 0 1 0 1 1 1
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| <math>(( p , q , r ))\!</math>
 
|}
 
|}
  

Revision as of 01:15, 23 August 2009

Logical Graphs

Truth Tables


\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)

\(\mathcal{L}_1\)

\(\text{Decimal}\)

\(\mathcal{L}_2\)

\(\text{Binary}\)

\(\mathcal{L}_3\)

\(\text{Vector}\)

\(\mathcal{L}_4\)

\(\text{Cactus}\)

\(\mathcal{L}_5\)

\(\text{English}\)

\(\mathcal{L}_6\)

\(\text{Ordinary}\)

  \(p\colon\!\) \(1~1~0~0\!\)      
  \(q\colon\!\) \(1~0~1~0\!\)      

\(\begin{matrix} f_0 \\[4pt] f_1 \\[4pt] f_2 \\[4pt] f_3 \\[4pt] f_4 \\[4pt] f_5 \\[4pt] f_6 \\[4pt] f_7 \end{matrix}\)

\(\begin{matrix} f_{0000} \\[4pt] f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0011} \\[4pt] f_{0100} \\[4pt] f_{0101} \\[4pt] f_{0110} \\[4pt] f_{0111} \end{matrix}\)

\(\begin{matrix} 0~0~0~0 \\[4pt] 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~0~1~1 \\[4pt] 0~1~0~0 \\[4pt] 0~1~0~1 \\[4pt] 0~1~1~0 \\[4pt] 0~1~1~1 \end{matrix}\)

\(\begin{matrix} (~) \\[4pt] (p)(q) \\[4pt] (p)~q~ \\[4pt] (p)~~~ \\[4pt] ~p~(q) \\[4pt] ~~~(q) \\[4pt] (p,~q) \\[4pt] (p~~q) \end{matrix}\)

\(\begin{matrix} \text{false} \\[4pt] \text{neither}~ p ~\text{nor}~ q \\[4pt] q ~\text{without}~ p \\[4pt] \text{not}~ p \\[4pt] p ~\text{without}~ q \\[4pt] \text{not}~ q \\[4pt] p ~\text{not equal to}~ q \\[4pt] \text{not both}~ p ~\text{and}~ q \end{matrix}\)

\(\begin{matrix} 0 \\[4pt] \lnot p \land \lnot q \\[4pt] \lnot p \land q \\[4pt] \lnot p \\[4pt] p \land \lnot q \\[4pt] \lnot q \\[4pt] p \ne q \\[4pt] \lnot p \lor \lnot q \end{matrix}\)

\(\begin{matrix} f_8 \\[4pt] f_9 \\[4pt] f_{10} \\[4pt] f_{11} \\[4pt] f_{12} \\[4pt] f_{13} \\[4pt] f_{14} \\[4pt] f_{15} \end{matrix}\)

\(\begin{matrix} f_{1000} \\[4pt] f_{1001} \\[4pt] f_{1010} \\[4pt] f_{1011} \\[4pt] f_{1100} \\[4pt] f_{1101} \\[4pt] f_{1110} \\[4pt] f_{1111} \end{matrix}\)

\(\begin{matrix} 1~0~0~0 \\[4pt] 1~0~0~1 \\[4pt] 1~0~1~0 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~0 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \\[4pt] 1~1~1~1 \end{matrix}\)

\(\begin{matrix} ~~p~~q~~ \\[4pt] ((p,~q)) \\[4pt] ~~~~~q~~ \\[4pt] ~(p~(q)) \\[4pt] ~~p~~~~~ \\[4pt] ((p)~q)~ \\[4pt] ((p)(q)) \\[4pt] ((~)) \end{matrix}\)

\(\begin{matrix} p ~\text{and}~ q \\[4pt] p ~\text{equal to}~ q \\[4pt] q \\[4pt] \text{not}~ p ~\text{without}~ q \\[4pt] p \\[4pt] \text{not}~ q ~\text{without}~ p \\[4pt] p ~\text{or}~ q \\[4pt] \text{true} \end{matrix}\)

\(\begin{matrix} p \land q \\[4pt] p = q \\[4pt] q \\[4pt] p \Rightarrow q \\[4pt] p \\[4pt] p \Leftarrow q \\[4pt] p \lor q \\[4pt] 1 \end{matrix}\)


\(\text{Table 1.}~~\text{Logical Boundaries and Their Complements}\)
\(\mathcal{L}_1\) \(\mathcal{L}_2\) \(\mathcal{L}_3\) \(\mathcal{L}_4\)
Decimal Binary Sequential Parenthetical
  \(p =\!\) 1 1 1 1 0 0 0 0  
  \(q =\!\) 1 1 0 0 1 1 0 0  
  \(r =\!\) 1 0 1 0 1 0 1 0  
\(f_{104}\!\) \(f_{01101000}\!\) 0 1 1 0 1 0 0 0 \(( p , q , r )\!\)
\(f_{148}\!\) \(f_{10010100}\!\) 1 0 0 1 0 1 0 0 \(( p , q , (r))\!\)
\(f_{146}\!\) \(f_{10010010}\!\) 1 0 0 1 0 0 1 0 \(( p , (q), r )\!\)
\(f_{97}\!\) \(f_{01100001}\!\) 0 1 1 0 0 0 0 1 \(( p , (q), (r))\!\)
\(f_{134}\!\) \(f_{10000110}\!\) 1 0 0 0 0 1 1 0 \(((p), q , r )\!\)
\(f_{73}\!\) \(f_{01001001}\!\) 0 1 0 0 1 0 0 1 \(((p), q , (r))\!\)
\(f_{41}\!\) \(f_{00101001}\!\) 0 0 1 0 1 0 0 1 \(((p), (q), r )\!\)
\(f_{22}\!\) \(f_{00010110}\!\) 0 0 0 1 0 1 1 0 \(((p), (q), (r))\!\)
\(f_{233}\!\) \(f_{11101001}\!\) 1 1 1 0 1 0 0 1 \((((p), (q), (r)))\!\)
\(f_{214}\!\) \(f_{11010110}\!\) 1 1 0 1 0 1 1 0 \((((p), (q), r ))\!\)
\(f_{182}\!\) \(f_{10110110}\!\) 1 0 1 1 0 1 1 0 \((((p), q , (r)))\!\)
\(f_{121}\!\) \(f_{01111001}\!\) 0 1 1 1 1 0 0 1 \((((p), q , r ))\!\)
\(f_{158}\!\) \(f_{10011110}\!\) 1 0 0 1 1 1 1 0 \((( p , (q), (r)))\!\)
\(f_{109}\!\) \(f_{01101101}\!\) 0 1 1 0 1 1 0 1 \((( p , (q), r ))\!\)
\(f_{107}\!\) \(f_{01101011}\!\) 0 1 1 0 1 0 1 1 \((( p , q , (r)))\!\)
\(f_{151}\!\) \(f_{10010111}\!\) 1 0 0 1 0 1 1 1 \((( p , q , r ))\!\)


Venn Diagrams

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